Abstract Introducing P T -symmetric generalized Scarf-II potentials into the three-coupled nonlinear Gross–Pitaevskii equations offers a new way to seek stable soliton states in quasi-one-dimensional spin-1 Bose–Einstein condensates. In scenarios where the spin-independent parameter c 0 and the spin-dependent parameter c 2 vary, we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross–Pitaevskii equations with P T -symmetric generalized Scarf-II potentials. We obtain analytical soliton states and find that simply modulating c 2 may change the analytical soliton states from unstable to stable. Additionally, we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations, exhibiting distinct behavior in energy exchange. For further investigation, we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components. These findings may contribute to a deeper understanding of solitons in Bose–Einstein condensates with P T -symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.
Abstract Introducing P T -symmetric generalized Scarf-II potentials into the three-coupled nonlinear Gross–Pitaevskii equations offers a new way to seek stable soliton states in quasi-one-dimensional spin-1 Bose–Einstein condensates. In scenarios where the spin-independent parameter c 0 and the spin-dependent parameter c 2 vary, we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross–Pitaevskii equations with P T -symmetric generalized Scarf-II potentials. We obtain analytical soliton states and find that simply modulating c 2 may change the analytical soliton states from unstable to stable. Additionally, we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations, exhibiting distinct behavior in energy exchange. For further investigation, we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components. These findings may contribute to a deeper understanding of solitons in Bose–Einstein condensates with P T -symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.
We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.
Abstract The dynamics of autonomous matter waves in two-component Bose–Einstein condensates subject to P T -symmetric potentials are investigated within the framework of coupled Gross–Pitaevskii equations. Families of nonlinear modes are obtained through analytical and numerical methods, and their stability properties are systematically characterized. The results demonstrate that P T -symmetric potentials play a crucial role in controlling the shape and interaction of matter waves, allowing for robust solitary-wave dynamics. The coexistence of a P T -symmetric Scarf-II potential and a δ -function potential further induces distinctive dynamical behaviors in multi-component condensates. These findings may provide a theoretical foundation for future experimental realizations and applications in autonomous matter-wave systems.
By introducing PT-symmetric δ-function potentials into three-component Gross–Pitaevskii equations that describe spinor F=1 Bose–Einstein condensates, we obtain stable and unstable analytical peakon solutions which enable us to explore the patterns of mean-field and spin-exchange interaction in relation to variations in energy of nonlinear modes. Furthermore, using iterative algorithms, we generate a series of numerical solutions and represent several families of peakon solutions in the form of energy curves, examining the impact of parameters on energy. Additionally, we observe a closed-loop structure in the family of peakons, with P0 and μ0 serving as the coordinate axes. On this curve, we discover stable peakons exhibiting periodic oscillatory properties, which can be regarded as a form of internal energy transfer within the coupled system. This research could contribute to a more comprehensive understanding of coupled nonlinear systems and serve as a reference for future experiments in this domain.
We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.
Recently, research of solitons in Bose–Einstein condensates has become a popular topic. Here, we mainly study exact analytical solutions of Gross–Pitaevskii equations describing spin-orbit coupled spin-1 Bose–Einstein condensates. To begin with, we show the analytical relation between different types of one-dimensional spin-orbit coupling and Zeeman effect. In addition, we find a transformation that can simplify the three-component Gross–Pitaevskii equations with spin-orbit coupling into the nonlinear Schrödinger equation. The abundant stripe phase and dynamic characteristics of the system are investigated.
We propose a method to analytically solve the one-dimensional coupled nonlinear Gross-Pitaevskii equations which govern the motion of the spinor Bose-Einstein condensates. In a uniform external potential, the Hamiltonian comprises the kinetic energy, the linear and the quadratic Zeeman energies. Several classes of exact periodic and solitonic solutions, either in real or in complex forms, are obtained for both the F=1 and F=2 condensates. These solutions are general that contain neither approximations nor constraints on the system parameters.